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Alejandro is selling HDMI cables on eBay, and is trying to determine the best price to sell at: For the last 10 weeks, he has adjusted his price slightly each week ...

Question

Alejandro is selling HDMI cables on eBay, and is trying to determine the best price to sell at: For the last 10 weeks, he has adjusted his price slightly each week and tracked the number of cables he sold: He plotted the results, and drew a line he feels fits the data well:3002501 200 I 150 100503.253.53.75 Price4.254.54.75a) The line of best fit passes through the points (3.50, 290) and (4.75,250). Find an equation for the line_Use variables: p for price in dollars, and Q for quantity of cables

Alejandro is selling HDMI cables on eBay, and is trying to determine the best price to sell at: For the last 10 weeks, he has adjusted his price slightly each week and tracked the number of cables he sold: He plotted the results, and drew a line he feels fits the data well: 300 250 1 200 I 150 100 50 3.25 3.5 3.75 Price 4.25 4.5 4.75 a) The line of best fit passes through the points (3.50, 290) and (4.75,250). Find an equation for the line_ Use variables: p for price in dollars, and Q for quantity of cables sold. Using this model, predict the number of cables Alejandro would sell at a price of $3.65,to the nearest whole cable. cables



Answers

At a concession stand at a high school football game, the owner notices that the relationship between the price of a hot dog and the number of hot dogs sold is linear. If the price is $\$ 2.50$ per hot dog, then approximately 650 are sold each night. If the price is raised to $\$ 3.50$, then the number sold drops to 475 per night. a. Make a graph with the price of hot dogs on the $x$ -axis and the number of hot dogs sold on the $y$ -axis. Use the points $(2.50,650)$ and $(3.50,475) .$ Then graph the line through the points with $x \geq 0$ b. Find an equation of the line through the points. Write the equation in slope-intercept form. c. Use the equation from part (b) to predict the number of hot dogs that would sell if the price were raised to $\$ 4.00 .$ Round to the nearest whole unit. (GRAPH CANT COPY)

In 39. In this case, we're going. Some information are it is known for you know itself. So word. The price for unit is $200. It was also the fire Turk. If he increased the price by $10 then the quantities which are sore, decreases by five units. What this means is that she engine price with respect to the change in foreign treaty is 10 or what? Minus four U minus people. That was chickpea services mainstream, actually no es saying darkened party X quantities are sword. So first, we need to find out how many quantities less than 1 50 years old and help us. So let's rewrite this as less Arugam 15. Subtract 1 50 that can beat it. And as 1 50 minus 1 15 minus six. So from this equation is cleared are 1 50 minus eggs units are less or because there's a minus sign. It means that the prize will increase by from 200. It will increase by since the sister increased factor and the more used minus here because the praises and freezing. So there will be two games, uh, changing the unit, which is one for minus six. So this is the 1 50 minus six. So the price eventually comes out us. This is 500 minus. Because their relation BMX No. One party, we have to find the maximum revenues revenue. We know that this is extreme speed and peace. He's already known that will be exchanged 500 minus to explicitly B minus two x two square pass by under. Next again. This is a parable opening downwards. Nastase maximum value when x is the vortex, which is Maiga's be worked through a So this would be minus off 500 times minus two, which is 1 25. This is the required value off expresses the vortex. So from X, we have to find our partisan revenue on what is the level should be the price we have between the price surprises 500 men stewards Price will be 500 minus two times expertise. When 25 this is nothing about 2 50. And here we have a daughter. So the price at which we should have we would have a maximum revenues to 50. The maximum revenue will be priced times quantity. So do 50 James one really favor that would be the maximum revenue and this comes out as 31,000 250. So this is the maximum revenue these are required answers.

Vilifying problem. We're looking at a power and light company Um that has a power plant on the river or where the river is 800 ft wide. And we see that there's different costs to lie it, a cable across the river and hawaii on land. And we want to determine the least expensive location. So we want to write a cost function. Well, we know that there's going to be 180, so CFX is going to be 180 times something, The last 100 times something. This will be the relationship of putting it over the water, which that's going to be um using Pythagorean here, um where we're going to have the change in the expert and the change in the apartment and then here to go on the land. It's just going to be in one direction. So that's just going to be the distance and feet in the extraction. So if we can relate the X and all the exes together, that way, we'll be able to find the cost, will be able to find a way to minimize cost

In order to go from power plant down here on day transfer up to the city have been bringing at the start here. We're going to follow that red line and noticing that part of it is gonna be along the river and that's going to be more expensive. And then part of it is along the land. We're going to come up with a cost function, so we'll go ahead and call that Sea of X here. And in order to generate that, we first want to ride out. Um, it's going to cost 180 ft in order to go along the river or across the river, we could say, and then it's going to cost $100 to go along the land. So we're going to come up with, ah, function relating these and also as a basis we're going to use this length acts over here on the diagram. So that first red line here is going to be the, um hi pot news of a right triangle. And because that's going to be along the river, we're gonna go ahead and multiply that by 180. And that length is the square root of Expert Plus 800 square because that's just again Pythagorean theorem, and that is the distance. And then we multiply that distance times 180 for the cost for foot and then plus, then we could multiply the 100 times the cost to get this distance and notice that that's just we convert two miles, which is 5280 times, too, and that gets us in the feet. And then we could subtract out that 10,560 ft we could subtract the X, and that would leave us with the circle part we're interested in. So let's go ahead and write that. So that would end up being 10,000 500 60 and then all of that minus X. And so this is our cost function. We could just rewrite it a little bit. Um, there's not much we could simplify but the first part there. But of course, you could expand out the 800 squared if you want to, but that would give us 180 times route and then leave that square about 700 spirit and then we have plus distribute that 100. And so that would just give us one million 56 1000 and then finally minus 100 times X. And there we go. So this is our cost function for part day. And we just took that step by step. And then for part B, we want to generate a table of values to find out if it's more cost effective to G o greater than or less than 2000. And so this is where we're just gonna go ahead and generate. Ah, lot of values here. Such Aziz, we're going to start with all the ones that are less than 2000. So we draw that. Well, that's fine. Just didn't wanna deal that for about. Okay, so like that. So let's start off with 535. So if that distance backs up there in the diagram that Z just to highlight that again, So that distance here, if that is 5. 35 then the cost is going to be one million 175,000 733 point Oh, sex. Okay, what if the cost is a little bit greater, like 1000 effects is 1000 on the cost would be closer to 1000 186 1000. 11 million, 106,512 and then 0.47 and the mid We'll just do one more here. Um, what if it's something like 15 and 50 for instance, that would end up being even more expensive. And so that would end up being the cost of one million, 214,000. So it looks so far like the cost is minimized when the length is less than 2000 right now, because it looks like guess we're approaching 2000. This way it looks like the cost is just getting bigger and bigger as we go. That direction. And so finally, we could just verify that. Whoops, Times that'll go all the way down. Sorry about that. Um, and then we're going to go ahead and generate. See school about that faster. Okay, so then we're going to go ahead and generate a table of values. Let's actually just do it up top here. So that way, why not? And ending up scrolling all the way down again. So let's do that. Not create that table. So what if you have the facts and our cfx? And again we generated these from the CFX function that we just found in part a there. So what if excess just a little bit bigger, like 2500 and 25. Thank Well, in that case, the cost would be 1000 one million, 280,000. Okay, so it is following this trend. It is getting more and more expensive. And that makes sense. Because as the amount of pipe that we're using to transfer across the river gets longer, then the cost is going to go way up because it's a lot more expensive to do that. And then let's test something in a little bit bigger than that 3500, and that would end up giving us a cost of about one million, 352,000. So we're really pretty confident at this point that Sorry. That's a five. There looked weird. Um, sorry. Just concentrating on getting that value right? Okay, um, we're very confident now that no matter what the length is that it is definitely going to minimize the cost. If it is less than 2000. So let's put that in here. 2000 ft. The party. But we'll go ahead and finish with this last value. Uh, if X is 5650 the cost would be one million 518,000 144 0.9 Okay, so that pretty much sums it up there. The cost was much, much smaller for something even less than 1000 ft. And so if we make this distance here pretty too small, then we're minimizing the amount of pipe that it took to get across. And then that allows us to spend less money overall. Because that distance there it is more cost effective toe minimize that length across the river.

Okay, so here for part A um we have the linear cost function represents, the cost is going to be cfx and that is going to get to produce X. X necklaces during a one month period. That's going to be equal to just 3.5 times X. And then plus 604 year. Mhm. All right. So there we have our function C of X. And for part B it is then given that we are going to sell the necklace for $25 each. So therefore the revenue is just going to be 25 times X. So our X is going to be equal to 25 X. And then for part C we have the r minus C. Of X. O. R minus C of X is equal to our of X minus C. Of Excel. Therefore it's going to be equal to 25 X minus 3.5 X plus 6 40. Which then is going to evaluate here to 21 0.5 X 25. X minus 2.5 X. And then the minus 6 40 so minus 6 40. And therefore here r minus C. F. X. Is the profit such that the cost is subtracted from the revenue. So ar minus cfx here represents the profit for selling necklaces. So all the profit. Mhm. Mhm. For selling mixes. Yeah And for parity. Um well the profit then if the artist sells 212 necklaces is just substituting in 212 into 21.5 X -640. So we have 21.5 times 212 And then -640 and that's going to give us 3,918. Therefore the profit for selling 212 necklaces is going to be $3,918. Take care. Yeah.


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